Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

rel="nofollow" href="#fb3_img_img_18f4e62d-6a9e-5be2-b5e4-668b872a6e88.png" alt="t element-of left-bracket a comma b right-parenthesis"/>, f Subscript n Baseline left-parenthesis t right-parenthesis equals f left-parenthesis t Superscript plus Baseline right-parenthesis

f Subscript n Baseline left-parenthesis t right-parenthesis equals f left-parenthesis t Superscript plus Baseline right-parenthesis

, we obtain

parallel-to f left-parenthesis t right-parenthesis minus f left-parenthesis t Superscript plus Baseline right-parenthesis parallel-to less-than-or-slanted-equals parallel-to f left-parenthesis t right-parenthesis minus f Subscript n Baseline left-parenthesis t right-parenthesis parallel-to plus parallel-to f Subscript n Baseline left-parenthesis t Superscript plus Baseline right-parenthesis minus f left-parenthesis t Superscript plus Baseline right-parenthesis parallel-to less-than-or-slanted-equals 2 v a r Subscript a Superscript b Baseline left-parenthesis f minus f Subscript n Baseline right-parenthesis comma

which tends to zero as n right-arrow infinity . Hence, f element-of upper B upper V Subscript a Superscript plus Baseline left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis .

We end this section with the Helly's choice principle for Banach space‐valued functions due to C. S. Hönig. See [127, Theorem I.5.8].

Theorem 1.34 (Theorem of Helly): Let be a BT and consider a sequence of elements of , with , for all , and such that there exists , with for all and all . Then, and . Moreover, if , with , for all , then and .

Proof. Consider a division d colon a equals t 0 less-than t 1 less-than midline-horizontal-ellipsis less-than t Subscript StartAbsoluteValue d EndAbsoluteValue Baseline equals b and let y Subscript i Baseline element-of upper F , with parallel-to y Subscript i Baseline parallel-to less-than-or-slanted-equals 1 , for i equals 1 comma 2 comma ellipsis comma StartAbsoluteValue d EndAbsoluteValue . Then, for all n element-of double-struck upper N , we have

StartLayout 1st Row 1st Column vertical-bar vertical-bar vertical-bar vertical-bar sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d times times left-bracket right-bracket minus minus of alpha alpha left-parenthesis right-parenthesis ti of alpha alpha left-parenthesis right-parenthesis t minus minus i 1 yi less-than-or-slanted-equals 2nd Column vertical-bar vertical-bar vertical-bar vertical-bar sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d times times left-bracket right-bracket minus minus of alpha alpha n left-parenthesis right-parenthesis ti of alpha alpha n left-parenthesis right-parenthesis t minus minus i 1 yi 2nd Row 1st Column Blank 2nd Column plus vertical-bar vertical-bar vertical-bar vertical-bar minus minus sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d times times left-bracket right-bracket minus minus of alpha alpha n left-parenthesis right-parenthesis ti of alpha alpha left-parenthesis right-parenthesis tiyi times times left-bracket right-bracket minus minus of alpha alpha n left-parenthesis right-parenthesis t minus minus i 1 of alpha alpha left-parenthesis right-parenthesis t minus minus i 1 yi comma EndLayout

where the first member on the right‐hand side of the inequality is smaller than upper M , since upper S upper B left-parenthesis alpha Subscript n Baseline right-parenthesis less-than-or-slanted-equals upper M . Moreover, by hypothesis, given epsilon greater-than 0 , there is upper N greater-than 0 such that, for all n greater-than-or-slanted-equals upper N , n element-of double-struck upper N , vertical-bar vertical-bar vertical-bar vertical-bar times times left-bracket right-bracket minus minus of alpha alpha n left-parenthesis right-parenthesis ti of alpha alpha left-parenthesis right-parenthesis tiyi Ì² less-than-or-slanted-equals StartFraction epsilon Over 2 StartAbsoluteValue d EndAbsoluteValue EndFraction comma for ModifyingBelow i With Ì² equals i comma i minus 1 period Hence, for all n greater-than-or-slanted-equals upper N ,

vertical-bar vertical-bar vertical-bar vertical-bar sigma-summation sigma-summation equals equals i 1 vertical-bar vertical-bar d times times left-bracket right-bracket minus minus of alpha alpha left-parenthesis right-parenthesis ti of alpha alpha left-parenthesis right-parenthesis t minus minus i 1 yi less-than-or-slanted-equals upper M plus epsilon

and we conclude the proof of the first part. The second part follows analogously.

For more details about functions of bounded variation, the reader may want to consult [68], for instance.

1.3 Kurzweil and Henstock Vector Integrals

Throughout this section, we consider functions alpha colon left-bracket a comma b right-bracket right-arrow upper L left-parenthesis upper X comma upper Y right-parenthesis and f colon left-bracket a comma b right-bracket right-arrow upper X , where upper X and upper Y are Banach spaces.

1.3.1 Definitions

We start by recalling some definitions of vector integrals in the sense of J. Kurzweil and R. Henstock. At first, we need some auxiliary concepts, namely tagged division, gauge, and delta ‐fine tagged division.

Definition 1.35: Let left-bracket a comma b right-bracket be a compact interval.

1 Any set of point‐interval pairs such that and for , is called a tagged division of . In this case, we write , where denotes the set of all tagged divisions of .

2 Any subset of a tagged division of is a tagged partial division of and, in this case, we write .

3 A gauge on a set is any function . Given a gauge on , we say that is a ‐fine tagged partial division, whenever for , that is,whenever

Before presenting the definition of any integral

Скачать книгу